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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The level sets of the moduli of functions of bounded characteristic

Author: Robert D. Berman
Journal: Trans. Amer. Math. Soc. 281 (1984), 725-744
MSC: Primary 30D50; Secondary 30D30
MathSciNet review: 722771
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Abstract: For $ f$ a nonconstant meromorphic function on $ \Delta = \{ \vert z\vert < 1\} $ and $ r \in (\inf \vert f\vert,\sup \vert f\vert)$, let $ \mathcal{L}(f,r) = \{ z \in \Delta :\vert f(z)\vert = r\} $. In this paper, we study the components of $ \Delta \backslash \mathcal{L}(f,r)$ along with the level sets $ \mathcal{L}(f,r)$. Our results include the following: If $ f$ is an outer function and $ \Omega $ a component of $ \Delta \backslash \mathcal{L}(f,r)$, then $ \Omega $ is a simply-connected Jordan region for which $ ({\text{fr}}\;\Omega ) \cap \{ \vert z\vert = 1\} $ has positive measure. If $ f$ and $ g$ are inner functions with $ \mathcal{L}\,(f,r) = \mathcal{L}\,(g,s)$, then $ g = \eta {f^\alpha }$, where $ \vert\eta \vert = 1$ and $ \alpha > 0$. When $ g$ is an arbitrary meromorphic function, the equality of two pairs of level sets implies that $ g = c{f^\alpha }$, where $ c \ne 0$ and $ \alpha \in ( - \infty ,\infty )$. In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.

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Keywords: Level sets, bounded characteristic, inner functions, Smirnov class
Article copyright: © Copyright 1984 American Mathematical Society